COMMISSION GEOLOGIOUE - Arkisto.gsf.fi

108 Bulletin de la Commission geologique de Finlande N: 0 212.
slowed down. That is to say, there will be a phase shift between the weighted
mean velo city of the medium and the velo city of the pebble, and the amplitude
of the motion will be smaller. Also the rotational component will be
retarded and the amplitude smaller. In this case, however, we have a contribution
to the lifting forces. This can be seen as follows (see Fig. 1). At a
specific instant, the sand and the pebble move to the right; at the same
time the elongation of both is positive. Then the speed of the pebble is in
general greated than that of the sand. Therefore, the pressure in front of
the pebble is larger than that of the rear side. When the elongation is positive,
then the rotational direction is clockwise. Therefore, the lifting force
generated by the friction at the high pressure side outweighs the negative
force at the low pressure side. From similar conclusions it can be deduced
that there will be a positive net contribution to the lifting force in this case.
In places, the sand has a tendency to become detached from the surface
of the pebble because of the relative motion of the sand and pebble so that
the sand grains will fall more or less freely. This causes a force because the
sand grains remain in the gaps created after the direction of the movement
has changed. This force is of importance at longer periods only, when the
particles have time to fall down. One force, which depends on the difference
of the density of the pebble and its surroundings, must also be considered.
Naturally, it can act in either direction. If we assume that the density of
the pebble and that of the medium are the same, this force is zero.
According to equation (10)
A = 2n V2v/a.
The wave length of the transversal wave motion in sand decreases when the
frequency increases. That is to say, when the frequency is low enough, the
wave length A is large compared to the diameter of the pebble. With higher
frequencies the wave legth may be of the same magnitude as, or small
compared to, the pebble size.
In the case of a small wave length, compared to the pebble size, the
above considerations about the lifting force partly lose their effect because
the amplitudes of the linear and rotational motions diminish considerably
with increasing frequency. Furthermore, it is no more evident that the
phase shifts are such that lifting forces will be created.
For low frequencies and long wave lengths, the relative velocities around
the pebble will be comparatively large. For, neglecting the influence of the
eosine factor, the maximum difference of the velocities just above and below
the pebble is, according to equation (10),
~ Vmn = -%z {a1' exp [- va/(2v)z J} D